University of Maryland Physics Education Research Group

Alternative Homework Assignment: Roller Coaster

Alternative Homework Assignment: Roller Coaster

Designing a roller coaster requires careful consideration of several
physics concepts. Most notably, mechanical energy must be kept track of
to ensure the Roller Coaster completes the course. Friction is also an
important consideration, however it is rather complicated to deal with
exactly. A simple but poor approximation can be made by assuming a constant
drag force resulting in a constant energy dissipation per unit track length.
This is not very accurate as shall be seen later, but it is better than
ignoring friction all together.

Another important design consideration is the accelerations experienced
by the passengers. The goal here is to maximze the excitement of the ride
without actually endangering the passengers. An important figure to remember
here is that an average person will black out if the "headward"
acceleration s/he experiences reaches 2 g. By comparison,
people can withstand accelerations on the order of 10 g in directions perpendicular
to their bodies without any permanent adverse effects, however most Roller
Coaster cars would jump their tracks at such extreme accelerations.

Part 1:

a) The object of this problem is to design a roller coaster with at
least the following features. 1) A large hill with a chain lift on one
side that is the source of energy. 2) A vertical loop or "loop the
loop". 3) A horizontal loop. 4) A series of at least three small hills.
You may include more features in your roller coaster if you wish. Begin
your roller coaster design by choosing the ordering of the features and
use your experience with roller coasters to make some rough estimates of
the dimensions of the features.

b) Now make the plan a little more exact. On graph paper, make a scaled
drawing of the track from the side in such a way that if you connected
the left side of the paper to the right to make a cylinder, you would have
a reasonable three dimensional picture of the track. In other words, even
though the track makes a large loop in reality, draw it as if the train
started on one end of the paper and ended on the other. Note that you will
need to take special care at the horizontal loop(s) to be sure the length
of track proposed is sufficient for the size loop(s) you intend. Use this
drawing to calculate the total length of the track for your roller coaster.

c) What about the train for your passengers? Assuming it consists of
five cars with a capacity of four people per car, estimate the total mass
of the train and its length.

d) Next use your knowledge of physics to see which parts of your rough
design work, and what needs to be altered in the parts that don't. The
first check is to make sure that the train has enough energy to reach the
end of the track. The Roller Coaster is powered entirely by the gravitational
potential energy it receives in being pulled to the top of the first hill.
For this reason, a good first check of the Roller Coaster is to make sure
that the first hill is the tallest feature on the ride. If this is not
the case on your drawing, choose dimensions that fix it, but do not draw
a new picture yet.

Part 2:

a) We also need to estimate the energy dissipated by frictinal drag
on the train for each meter of track it traverses. As a lower limit on
this frictional energy loss let's consider one of the tallest and fastest
roller coasters in the world. It has a height of 62.5 m and a top speed
of 34 m/s. This speed is reached at the bottom of the first hill which
has a slope of 60. Comparing its actual speed at the bottom of this hill
to that achieved in the frictionless case, estimate the average energy
dissipated per meter of track going down this hill. Take this to ba a reasonable
minimum rate of energy dissipation.

b) At the end of the ride, the Roller Coaster will need to be stopped
without accelerating the passengers too violently. For this to be the case,
the ride will have to dissipate enough energy for the final speed to be
about five meters per second. Use this final speed, the height of your
track's first hill, and the total length of your track to estimate a maximum
rate of dissipation. Use the two rates you have calculated to make a reasonable
estimate of the rate of dissipation for your roller coaster.

c) With an estimate of the rate of dissipation now in hand, it's time
to check that your roller train can rally make it up to and over each of
the highest points on your roller coaster. Measure the amount of track
the train has to travel over to reach each of those points. From these
data, calculate the amount of energy the train has at each of these points.
If you find that there is any place where the train is higher than the
available energy allows, then rescale, and/or reorder the features so that
the train is able to reach every point on the track.

Part 3:

a) Now consider the vertical loop in some detail. In order to stay on
the track without depending on the wheels beneath the track, the train
must experience a downward, centripetal acceleration at the top of the
loop that exceeds 1 g. The train's speed is determined by its height and
the amount of track it has traveled across. Considering this, what should
the maximum radius of your track's vertical loop be.

b) In the frictional case, the point were the train is most likely to
fall is not the top of the loop, but the point where the component of the
train's weight in the direction of travel is equal in magnitude to the
frictional force. Why is this the case, and where is this point on your
track? What should the new radius of the loop be?

b') (optional) The above is a case where the constant friction approximation
is particularly poor. If the air friction is considered to be negligible,
then the frictional force should be proportional to the normal force. The
normal force in the loop is a function of the train's position and speed.
If the average friction used above is the frictional force on level ground,
then where is the new critical point?

c) The above analysis is fine for the train's center of mass, but is
not accurate for the first or last car. The reason for this is that the
entire train must always have the same speed which is determined by the
center of mass. The passengers in the middle car experience essentially
this motion. The passengers in the front and back however have a slightly
different experience. Calculate the train's speeds when the first and last
cars are at the top of the loop. From this calculate the acceleration felt
by the first and last car's passengers. What should the loop radius be
if all of the cars are to stay on the track? (If you really wanted to treat
the friction accurately in this problem you would have to consider that
each car has its own normal force and that all of them contribute to the
overall friction.)

d) Calculate the acceleration experienced by the passengers of the first,
middle and last cars at the bottom of the loop. Is this reasonable?

e) If the bottom of the loop is causing your passengers to black out,
the acceleration may be reduced by cutting the circle of the loop into
four quarters and replacing the bottom two quarters with sections of a
much larger circle. Design such a loop that takes the track from horizontal
back to horizontal without causing any of the passengers to black out.
Make sure the train will still stay on the track.

Part 4:

a) Consider the other features of your Roller Coaster one by one for
safety. The best way to do this is to find the place with the greatest
acceleration and make sure that it is safe for the first, last, and middle
cars.

b) As a last touch, find the slope of the bank for the horizontal loop.
You can do this by making sure the force on the passengers is always perpendicular
to the track. If your loop dimensions allow it, have the track sloped so
as to keep the velocity constant throughout the horizontal loop. This should
make your calculations easier.

c) Use all of the adjustments you have made to make a final draft of
your Roller Coaster drawing. Make sure that the train can reach all of
the critical points with appropriate velocities. Calculate the maximum
and final velocities speeds of the train as it completes a cycle.

d) Explain what car you would prefer to ride in and why. (The answer
to this might depend greatly upon your personal feelings about roller coasters.)

"g" as a unit of acceleration refers to the gravitational
acceleration near the surface of the earth (9.8 m/s2). This unit is often
used when referring to accelerations experienced by a person in a vehicle
because when the vehicle accelerates, it pushes the passenger along with
it. This pushing changes the normal force experienced by the passenger
and the person feels as if s/he is in a gravitational field that is the
resultant of the actual gravitation field and the negative of the vehicle's
acceleration. For example if the vehicle were accelerating in the up direction
at 1g, the passenger would experience 2g of acceleration and pass out while
if the vehicle were accelerating down, the passenger would feel weightless.

These problems written and collected by K. Vick, E. Redish, and P. Cooney.
These problems may be freely used in classrooms. They may be copied and
cited in published work if the Activity-Based Physics (ABP) Alternative
Homework Assignments (AHAs) Problem site is mentioned and the
URL given.